Papers

We prove finite-sample risk bounds for empirical quadratic and power Fréchet means in geodesic metric spaces with curvature bounded above for observations with arbitrary joint law. We compare the sample with exchangeable copies in which one observation is redrawn independently of the original sample and the others move to preserve the joint law. These displacements control the error, with no entropy or dimension assumptions. For quadratic Fréchet means, this yields a general “coupling bound” in any separable complete $CAT(\kappa)$ space, where for $\kappa>0$ the observations must lie in a closed convex set of circumradius below $\pi/(2\sqrt{\kappa})$. This bound is exact on the real line for nonnegatively correlated Gaussian observations and for comonotone functions of a common factor. We apply it in two settings. Under local dependence, the resulting risk bound grows linearly with the average size of the dependency neighborhoods. Under transport sensitivity and for bounded metric variances, summable sensitivities of the conditional laws to a change in one observation preserve the parametric mean squared error rate, while nonsummable polynomial decay gives explicit slower rates. We verify transport sensitivity for stationary reversible Markov chains with Wasserstein-contractive kernels and for bi-Lipschitz images of Gaussian sequences. In the more challenging setting of heavy-tailed, dependent data with possibly infinite variance, we consider power Fréchet means of order $\alpha\in(1,2)$. Under nonpositive curvature, we bound their $L^\alpha$ risk assuming only finite $\alpha$-th moments.

In this paper we propose and study arctan-skewing, a general approach to skew any symmetric distribution on the unit hypertorus. Arctan-skewing retains the main advantages of the popular sine-skewing mechanism, but improves on it by overcoming its two major limitations. These are a condition on the skewness parameter and a singular Fisher Information matrix in the vicinity of symmetry when combined with well-known symmetric models from the literature such as the circular von Mises or bivariate Cosine models. Arctan-skewing thus represents a compelling alternative to model skewed data on the circle and torus, notably from the field of protein bioinformatics. After a thorough investigation of its stochastic properties, we present a maximum likelihood estimation procedure and devise optimal tests for symmetry against our new class of skewed alternatives. Finally we show that arctan-skewed models better fit protein data compared to their sine-skewed counterparts.

Threshold and change-plane models provide a useful framework for capturing regime-dependent dynamics, but most existing approaches typically rely on scalar thresholds or linear splitting boundaries. We propose a scalable semiparametric change-plane regression framework that allows the regime boundary to depend nonlinearly on multivariate state variables. Random Fourier features provide a flexible and computationally efficient approximation to the nonlinear boundary function. A two-stage procedure first estimates the induced regime classification using a smooth surrogate and then refits the structural parameters by hard-threshold OLS. We establish convergence of the first-stage regime classification and show that its estimation error is asymptotically negligible for the second-stage refit, yielding oracle equivalence. We further develop a dependent-multiplier bootstrap Sup-Wald test for detecting the existence of regime effects when the boundary is unidentified under the null, and establish its asymptotic validity. Simulations demonstrate accurate regime recovery, near-oracle performance of the proposed estimator, and satisfactory finite-sample behavior of the Sup-Wald test. An application to postwar U.S. unemployment data provides strong evidence of regime dependence, while the estimated nonlinear change plane identifies a distinct stress regime characterized by significant shifts in short-run unemployment dynamics.

Background: Credit risk modelling is fundamental in financial institutions, typically using credit history, repayment behaviour, income, and outstanding debt to estimate a borrower's probability of default. Although transactional data are sometimes incorporated through aggregated measures such as recency, frequency, and monetary indicators, behavioural information embedded in raw transaction narratives remains largely unexplored due to its unstructured nature. This limits opportunities to extract richer behavioural insights. Aim: This research investigates payment narratives from transactional data as an alternative data source for credit risk modelling. Setting: The study uses transactional data from a large South African financial institution focused on small and medium enterprises. Method: Techniques such as natural language processing and regular expressions are used to transform unstructured transaction descriptions into meaningful, explainable behavioural features by categorising transactions into interpretable spending categories. A novel categorisation algorithm was developed and evaluated for classification accuracy. The extracted features are incorporated into a credit risk scorecard to assess their predictive value. Results: The categorisation algorithm demonstrates high classification accuracy. Using these features in the scorecard resulted in a significant increase in Gini compared to a scorecard without them. Conclusion: Text-based transactional information can be systematically extracted, categorised, and operationalised to enhance credit risk model performance. Contribution: The study develops a novel, integrated, explainable methodology for transforming transactional text into behavioural features, introduces a structured RegEx-based categorisation approach, and demonstrates their predictive value within traditional scorecard frameworks.

Dynamic Time Warping (DTW) is a common choice for measuring the discrepancy between time series. It proceeds by finding an optimal warping between timestamps through dynamic programming, which can be slow and is not differentiable. Consequently, several variants have been proposed. Among them, Soft-DTW proposes a smooth formulation of DTW, amenable to gradient computation, while Continuous-DTW (CDTW) introduces a continuous formulation through parametrizable warping functions. Yet, computational costs remain high for both methods and impractical for large datasets in modern machine learning pipelines. This paper revisits CDTW in several ways. After establishing its differentiability under mild conditions, we introduce FastCDTW, which is an efficient approximate solver with two different computational strategies: an exact closed-form scheme for piecewise-linear warping functions, and an unbiased Monte Carlo estimator for smooth ones, such as implicit neural networks. Both strategies provide gradients with respect to the warping function and the input time series, and the resulting algorithm is GPU-friendly. We demonstrate its benefits through its integration into more complex learning problems, including time series averaging, classification and deep domain adaptation.

A solid understanding of the predictive success of deep neural networks (DNNs) remains elusive. For example, while many data-dependent metrics exist that seek to predict model quality of DNNs, these metrics may be expensive to compute for large datasets and/or may be impossible to compute when training datasets are not released. A practical and effective predictor of model quality involves analyzing DNN weight matrices, as prior work has established that heavy-tailed spectral distributions in trained weight matrices can be strongly predictive of model quality. However, such metrics lack a rigorous statistical justification. In this work, we aim to understand why these metrics perform well, from a statistical perspective. To do so, we employ a probabilistic framework to derive the marginal likelihood for a DNN, evaluated using only the final layer weights. Across large-scale convolutional DNNs and LLMs, we find that this marginal likelihood predicts model quality. To understand the role heavy-tailed spectra play in predictive performance, we derive the limiting log-marginal likelihood for spectra that follow the (heavy-tailed) High-Temperature Marchenko-Pastur (HTMP) distribution. We show that this limiting quantity is convex in a heaviness parameter, and we derive an optimal heaviness that empirically predicts over-fitting for large-scale DNNs.

Random spanning forests provide probability models for partitions and Monte Carlo methods on graphs. In these applications, the distribution produced by a sampler determines which forest functionals can be estimated. The Aldous–Broder algorithm constructs a spanning tree from the first-entrance edges of a covering random walk. Its distribution is known, but a direct calculation requires summing over all orders in which a fixed tree can be discovered. We evaluate this sum through an identity for rooted forests and real matrices with zero row sums. Under a nonsingularity condition on the principal submatrices encountered during growth, the sum of products of harmonic extension entries equals a principal cofactor of a matrix on the roots. With one root, the sum is one. The identity gives a direct proof of the extended Aldous–Broder theorem for every finite irreducible Markov chain started at a fixed state, without assuming reversibility or bidirectional transition support. It also gives the conditional distribution of the discovery order and the law of the forest obtained by stopping after all nonroot vertices have been visited. A completion weight describes the difference between this stopped-walk law and the weight-proportional forest law, including the support restriction needed for importance correction. Numerical examples illustrate the sampling implications. The identity, the Aldous–Broder theorem, and the stopped-forest law are formalized in Lean4.

Trajectory extrapolation from a partial target trajectory is underdetermined, but completed related trajectories provide information about the unresolved continuation. We propose target-constrained pooling (TCP), which separates the finite-horizon endpoint from continuation shape and estimates that endpoint from two complementary source signals. Uniform pooling captures final response levels, while matched-increment pooling captures the remaining change after aligning source and target states. TCP selects representative sources, projects both endpoint estimates onto a target-consistent region, converts them into corrections of the free forecast, and controls their influence through strength calibration and degradation-aware weighting. Across 25,245 targets from five domains and both forecasting directions, TCP reduces mean RMSE by 35.8% to 42.1% for all ten target-only forecasters and improves all 100 forecaster, dataset, and direction comparisons. The two endpoint signals consistently outperform either alone, while representative selection remains within 0.64% to 1.02% of the full donor pool using 12.7x fewer donor inputs and 138.8x fewer matching distance computations.

Simulation-based inference (SBI) enables Bayesian inference when a simulator is available but the likelihood is intractable. Modern SBI methods train neural networks on simulated data to obtain an amortized approximation of the posterior, making inference for new observations fast. A central challenge in SBI is to extract informative, low-dimensional summary statistics; without them, performance can degrade rapidly as the observation dimension grows. We therefore highlight summary learning as a critical component of SBI and propose a general framework that jointly learns a neural-network-parameterized summary and an amortized posterior estimator. We further introduce a local refinement strategy that adapts the amortized estimator to a specific observation, thereby reducing the amortization gap. Both amortized training and local refinement accommodate implicit posterior estimators without tractable densities. On the theoretical side, we analyze the finite-sample behavior of the joint learning procedure through the lens of nonparametric conditional distribution estimation, and show on representative examples how low-dimensional summaries can mitigate the classical curse of dimensionality. Empirically, our results support this motivation and demonstrate favorable performance relative to existing methods.

Effect sizes for single-case experimental designs usually rely on models for the baseline trend and the serial dependence, which are challenging to check with few sessions. We develop design-based effect sizes for multiple-baseline experiments with a randomized intervention onset. We define case-level and pooled standardized effects with potential outcomes and propose weighted estimators that are unbiased or asymptotically unbiased without an outcome model. The usual difference between phase means is generally biased when the untreated outcome changes over sessions. For research synthesis, we provide variance estimators and asymptotic confidence intervals that require no model for serial dependence, and variance formulas for planning. Simulations and three randomized studies of reading and writing interventions illustrate the methods.

If a second text corpus were collected from the same population, how much would a predictor's outputs change? We study this question for masked-text reconstruction using a smoothed predictor built from nearby token-pair counts and evaluated over the full 50,257-token GPT-2 vocabulary. Treating each source document as one observation, we derive a first-order formula for the expected disagreement between predictors fitted to two independent corpora. The formula accounts for variable document length, random context counts, and correlations across token offsets, and it can be evaluated from a separate reference corpus without access to either target corpus. On TinyStories, WikiText-103, and CNN/DailyMail, growing each corpus from 512 to 2,048 chunks of 64 tokens lowers disagreement by only 5–40%, far short of the 75% implied by $1/n$ scaling. The forecast tracks this slow decline, which we trace to smoothing of rarely seen contexts. Forecast accuracy depends less on how these correlations are modeled than on the size of the reference corpus: small references systematically underpredict disagreement. A source-split jackknife that uses only reference data reduces this calibration error by 57–59%. After this correction, a simpler smoothed-multinomial approximation is most accurate when the reference corpus is smaller than the target corpus, and a nonlinear source bootstrap is most accurate when it is larger. Finally, small masked diffusion language models trained on the same corpora show that the count-based forecasts partly identify which inputs are most sensitive to the corpus, but miss the neural models' much faster decline in disagreement.

Continuous-time latent state-space models are widely used for irregularly observed time series, typically representing uncertainty through a random initial state followed by smooth ordinary differential equation (ODE) dynamics, or Brownian perturbations accumulated by a stochastic differential equation (SDE). While these constructions are natural for smooth or diffusive systems, they do not directly encode a finite set of unobserved shocks whose timing, magnitude, and effect on memory are themselves uncertain. We instead introduce latent jump models (LJMs): the latent state follows closed-form deterministic dynamics between Poisson-driven event times and undergoes a learned instantaneous jump at each event, with state-dependent intensity and mark distributions. This defines a time-homogeneous Markov process whose jumps are intrinsic state transitions rather than numerical integration steps. For training, we derive a tractable event-augmented path-space evidence lower bound (ELBO) and introduce Scan-Parallel Variational Inference (SPVI): an observation-conditioned variational posterior samples the complete latent event configuration before the state recurrence is evaluated using triangular affine scans. This enables fast and scalable training for LJMs in logarithmic temporal depth. Across synthetic stochastic systems and real irregular time series, LJMs remain competitive with continuous-time latent baselines in forecast accuracy, often with fewer latent updates than numerical solver evaluations, while training one to two orders of magnitude faster than the sequential baselines.

Spatial prediction is a critically important task in data analyses that involve spatially or spatio-temporally-referenced data. It is commonly carried out using hierarchical models, which comprise a process model describing the latent process of interest, and a data model describing how the process is observed. The target quantity is the predictive distribution, which is the conditional distribution of the latent spatial process given the observed data. Unfortunately, the predictive distribution is rarely available in closed form and can be difficult to sample from. Here, we show how recent developments in guided diffusion modelling from machine learning can address a wide class of problems in spatial prediction when the likelihood function derived from the data model is known and differentiable, possibly after relaxation. To show the versatility of the approach, we apply it to both Gaussian and non-Gaussian process models and a variety of data models that describe direct partial observations, linear and areal aggregates, counts, right censoring, and area-exceedance functionals. Where possible, we validate the framework against exact and Markov-chain-Monte-Carlo (MCMC)-derived predictive distributions. We demonstrate our approach on several simulation studies and in a controlled experiment of methane-flux inversion over the UK and Ireland. We show that guided diffusion can sample from the correct predictive distribution in a variety of settings, that it can be applied to models where MCMC is intractable, and that it scales well with the dimensionality of the latent process.

Comparing distributions under severe sample imbalance requires effectively using abundant majority observations while retaining sensitivity to differences involving the smaller sample. This challenge is particularly relevant for complex, non-Euclidean data, such as graphs, whose structural characteristics call for flexible testing procedures. Classifier two-sample tests (C2STs) provide such flexibility, but existing procedures are primarily designed for balanced or comparably sized samples. In this paper, we study C2STs in a highly imbalanced regime, and show that severe sample imbalance may substantially degrade testing performance. Different from conventional imbalanced learning approaches that treat sample imbalance merely as a nuisance for classifier training, we turn it into a source of repeated testing evidence. Building on this insight, we propose an Enhanced Imbalance-Aware Deep Classifier Two-Sample Test (EIAD-C2ST), a classifier-agnostic framework designed for sample imbalance that accommodates diverse data modalities by tailoring classifiers to data structure. For computationally expensive deep classifiers, we further develop a scalable representation-amortized implementation strategy. Theoretically, we establish rigorous guarantees on asymptotic size and characterize conditions for power consistency, allowing the imbalance ratio to diverge. We demonstrate the empirical gains using simulations and an application to blockchain forensics. The results reveal that the majority-sample surplus, when properly leveraged, can be organized into useful statistical evidence, rather than an obstacle, for distributional comparisons with complex data.

Conformal prediction provides finite-sample, distribution-free coverage guarantees that can support decision-making in specific settings. In multistage problems, earlier decisions determine the populations relevant to later calibration. When later scores are refit using earlier calibration results, reusing those cases for ranking need not preserve exchangeability with a surviving test case. We study sequential conformal calibration with a fresh fold at each stage: conditional on earlier folds, the selected fold scores and the surviving test score are exchangeable. This yields finite-sample coverage guarantees for every prefix of the decision sequence. For atomless survivor score distributions, we derive a universal Binomial-thinned Beta chain for the realized coverage process. Its law depends only on fold sizes and target levels, even under predictable refitting, and yields exact coverage slack, abstention probabilities, and calibration-conditional guarantees. We further characterize the tradeoff between freshness and data reuse. For fixed scores, strict-survivor reuse is exactly valid. For adaptive scores refit using past calibration data, reusing survivors can incur worst-case rank loss, while blocked designs provide an intermediate option between full reuse and stagewise fresh folds. As a case study, we apply the framework to serial multi-echelon inventory problem and propose a sequential conditional-quantile formulation. Each stock level is learned as a context-dependent quantile of cumulative demand among scenarios satisfying earlier no-backlog constraints. We connect these targets to common-shift stationarity and derive decision-level implications for the resulting calibrated policies. In a three-echelon simulation, fresh-fold refitting achieves 81.98% final-prefix coverage against an 81.82% target while reducing mean cost by 5.5% relative to the Bonferroni baseline.

We study the eigenvector-weighted spectral statistics \[ T_n(\varphi,g)=p^{-1}tr\{\varphi(\boldsymbol{S}_n)g(\boldsymbol{\Sigma})\} \] of large-dimensional sample covariance matrices \(\boldsymbol{S}_n\) with population covariance \(\boldsymbol{\Sigma}\). These statistics combine functions of sample eigenvalues with sample–population eigenvector overlaps and arise in statistical losses that depend on both. As the dimension and sample size grow proportionally, we establish a joint central limit theorem for \(\{T_n(\varphi_a,g_a)\}_{a=1}^r\) with multiple analytic test functions and population weights, centered by finite-dimensional deterministic equivalents. A key intermediate step is a joint Gaussian process limit for the weighted resolvent traces \[ m_n^g(z)=p^{-1}tr\{(\boldsymbol{S}_n-z\boldsymbol{I}_p)^{-1}g(\boldsymbol{\Sigma})\}. \] The limiting means and cross-covariances are expressed in terms of the population spectral distribution. We apply these results in two settings: one concerns the variance component of minimum-norm prediction risk, and the other concerns the Frobenius loss of fixed rotationally invariant covariance estimators. For fixed affine shrinkage rules, the loss means, variances, and cross-covariances have explicit formulas in terms of population spectral moments.

We study dictionary learning for probability measure-valued data by embedding measures into Hilbert spaces using features derived from optimal transport geometry. We learn data-adapted dictionaries through sparse autoencoder parameterizations, producing fast, scalable representations of probability measures as sparse finite-dimensional codes. The resulting framework combines the rich, geometric structure of optimal transport with the interpretability and efficiency of sparse representation learning. We demonstrate its utility across diverse data modalities, including hyperspectral images, geometric point clouds, and document embeddings in large language model latent spaces. We also prove generalization bounds for sparse dictionary learning objectives on broad classes of $L^2$-valued measure embeddings, expressed in terms of the metric complexity of the atom class. For optimal transport and entropic transport map dictionaries, we derive new covering number bounds that yield explicit rates. Together, our computational and theoretical results support a new, scalable model for representation learning in the space of probability measures via embedding into Hilbert spaces.

We consider online binary prediction with paid stochastic features, where the learner can spend resources to obtain better observations of the underlying features. This setting is motivated by applications such as medical testing, where additional cost can provide more precise measurements of patient's features for predicting a binary outcome. At each round, a latent feature is drawn i.i.d., the response is generated through a logistic model, and the learner chooses how much to pay before observing a noisy version of the feature. The goal of learner is to jointly control prediction loss and acquisition cost. In contrast to the former result on the online linear regression with paid features, covariance information alone is no longer sufficient: two noise distributions with the same covariance may induce different optimal logistic losses, and lower variance can even lead to higher loss. Under a structural ordering in which larger payments yield better observations by removing independent noise, we obtain $\widetilde O(\sqrt{T})$ regret when the noise structure is known and $\widetilde O(T^{2/3})$ regret when it is unknown, with matching lower bounds up to logarithmic factors.

For Gaussian observations, a deterministic multiple of Hotelling's $T^2$ statistic has an exact $F$ distribution. This paper investigates when this approximation remains valid for an independent and identically distributed (i.i.d.) data matrix $\mathbf{X}\in \mathbb{R}^{p\times n}$ with a fixed, possibly heavy-tailed entry law $\xi$. Under symmetry and $\limsup_{n\rightarrow \infty} p/n<1$, we prove that approximation by the $F$ distribution in Kolmogorov distance is equivalent to belonging to the domain of attraction of the normal law. Both conditions are equivalent to relative concentration of the diagonal entries of the null-space projection $\mathbf{P}=\mathbf{I}_n-\mathbf{X}^{\top}(\mathbf{X}\mathbf{X}^{\top})^{-1}\mathbf{X}$. The central quantity $R=\mathbf{1}^{\top}\mathbf{P}\mathbf{1}$ measures the geometric alignment between the averaging direction and the random row space of $\mathbf{X}$. For symmetric finite-variance entries, the sufficient approximation extends to every sequence $1\le p<n$, including fixed nullity $n-p$. As a consequence of the $F$ approximation, Hotelling's $T^2$ statistic exhibits different asymptotic distributions in different regimes, including the normal, chi-square, and inverse-chi-square limits depending on $p$ and $n$. Without symmetry, we establish a central limit theorem under the weak-$5/2$ tail condition $\lim_{x\to\infty}x^{5/2}\mathbb{P}(|\xi|>x)=0$ when $p/n\rightarrow \gamma\in (0,1)$, and obtain the $F$ approximation as a consequence. Centered standardized Pareto laws show that the exponent $5/2$ is sharp for this universal absolute-tail sufficient condition, although balanced-tail nonsymmetric laws can retain the approximation at the boundary, which reveals that the asymmetry of the tails matters for the $F$ approximation.

Determinantal point processes (DPPs) model subset selection probability with repulsion, but many applications also exhibit attraction arising from pairwise relations or shared latent structures. Nonsymmetric DPPs (NDPPs) capture both effects, but their kernels are difficult to interpret and specify. We propose the determinant-ratio point process (DRPP), which models repulsion and attraction via a ratio of the determinants of two symmetric positive-definite kernels. This formulation preserves the feature-based interpretation of standard DPPs while allowing the two effects to be specified separately. Furthermore, we introduce an auxiliary-variable representation that enables efficient MCMC sampling and yields tractable expectation formulas for key statistics including the normalizing constant, which together facilitate maximum-likelihood learning. We theoretically characterize the interaction structures well suited to DRPPs: for clustered attraction with $C$ latent groups, DRPPs exactly realize the target pairwise interactions with an attraction rank $C$, whereas NDPPs can require a rank that grows with the number of items given the same repulsion component. Conversely, nontransitive pair-specific attraction may require compensation through the DRPP repulsion kernel. Experiments support these predictions, showing that DRPPs fit transitive attraction with substantially fewer parameters than NDPPs.

We study the asymptotic properties of the nonlinear stochastic GMM algorithm proposed by Chen, Lee, Liao, Seo, Shin and Song for recursively estimating parameters defined by possibly nonlinear and overidentified moment restrictions $\E[g(X,\beta^*)]=0$. Each observation updates estimates of the moment Jacobian, inverse covariance matrix and Gauss–Newton preconditioner at a cost independent of sample size. For linear and nonlinear instrumental-variable moments, we derive an exact rank-two Sherman–Morrison–Woodbury update. The algorithm combines preconditioned stochastic approximation with iterate averaging. Under a preconditioned monotonicity condition, satisfied globally in linear models and locally around a consistent pilot estimate in general, we establish almost sure convergence and rates, as well as a functional central limit theorem with the efficient GMM variance. These results require four moments of the moment function and its Jacobian when the step exponent exceeds $2/3$. The inverse Gauss–Newton matrix consistently estimates this variance, and a projected linearised statistic enables online testing of overidentifying restrictions. We extend the results to $L$-dependent data with long-run variance weights under stronger moment assumptions. Simulations show that the averaged estimator approaches offline two-step GMM in accuracy and coverage at $10^6$ observations, while requiring a fraction of its computing time and fixed memory at up to $10^7$ observations. The online test achieves correct size and comparable power. In an application estimating the effect of a third child on weeks worked using $254{,}654$ women from the 1980 US Census, single-pass estimates agree with offline GMM within a fraction of a standard error on average across data orderings, and both methods yield the same overidentification-test conclusion in most orderings.

Multivariate longitudinal count data arise in biological studies, where responses may be over-dispersed, correlated within subjects and over time, measured at irregular times, and exhibit unobserved subgroup structure. Our goal was to develop a model that can accommodate over-dispersion, longitudinal dependence, cross-outcome dependence, covariate effects, irregular observation times, and covariate-dependent latent-class membership. We propose a latent-class Poisson-lognormal mixed model for clustering heterogeneous multivariate count trajectories. Estimation is performed using a variational EM algorithm designed for scalable inference. Simulation studies show accurate parameter estimation, strong latent-class recovery, and reliable selection of the number of classes, as compared to state-of-the-art approaches. We also applied the proposed approach to longitudinal neutrophil RNA-seq data from hospitalized COVID-19 patients as an illustration. The proposed framework provides a flexible approach for clustering complex longitudinal biological count data. Implementation code is available at https://github.com/PengqiLiu-Mc/latent-class-mpln-mixed-models.

This work studies an entropy-based estimator of Kullback-Leibler (KL) divergence constructed from normalized importance weights. We prove a universal finite-sample bound showing that its expectation can exceed the target-proposal KL divergence by at most $\log 2$ for any sample size. We then characterize the KL divergence between the distribution induced by importance-weighted resampling and the proposal, showing that it is upper bounded by both the expectation of the entropy-based estimator and the target-proposal KL divergence. Importantly, these bounds require neither bounded importance weights nor finite higher moments. We apply these results to test-time alignment of generative models via Soft Best-of-$N$ sampling, an importance-sampling-based scaling technique proposed by Verdun et al. (2025), and to relative entropy coding, obtaining finite-sample policy-divergence certificates and moment-free bounds on coding cost.

Bayesian simulation-based inference (SBI) can struggle to recover increasingly concentrated posteriors as observation size grows. We consider learning sufficient summary statistics jointly with the posterior as a general principle for scalable inference in various SBI methods. A common measure taken in SBI is dimension reduction through predetermined summaries, which may partially discard posterior information. Our approach instead imposes structure on how the posterior learner depends on a learned summary (e.g., affinity), and learns the summary simultaneously with the posterior. We identify an important mechanism named two-way guidance: under a structural restriction on the summary, the posterior sufficiency of the learned summary simplifies posterior learning, whereas a suitable posterior objective guides the learned summary towards posterior sufficiency. This mechanism explains a counterintuitive finding that the structural restriction on the summary statistics improves posterior learning, rather than limit its performance. We establish theoretical guarantees on both posterior estimation error and the learned summary's departure from sufficiency, thus identifying conditions on the simulation budget for recovering increasingly concentrated posteriors as observation size grows.

Determinantal point processes (DPPs) on a finite set are probability models that exhibit repulsion where inter-item similarities are encoded by a positive definite kernel matrix. Scaling DPP learning to large datasets typically requires imposing structural constraints on the kernel matrix, such as a low-rank-plus-diagonal form. However, structural assumptions introduce a severe trade-off between expressive power and computational scalability. Our study addresses this challenge by (a) modeling the full-rank interactions via a nonlinear kernel function and latent features, (b) designing a surrogate function for the maximum likelihood estimator, and (c) adopting stochastic optimization using random features of the nonlinear kernel. Numerical experiments on real-world datasets demonstrate that our method significantly improves this trade-off, achieving better generalization performance with remarkable parameter efficiency.

Estimating (heterogeneous) treatment effects is still challenging when the outcome is semicontinuous (i.e., exhibiting both a discrete point mass at zero and a skewed continuous distribution for positive values), despite the strong presence of this data type in real world settings. Existing methods either ignore the zero mass, impose fragile parametric assumptions, or suffer from extreme computational costs and data augmentation noise. This paper proposes Zero-Inflated Continuous Bayesian Causal Forests with Duan's Smearing (ZIC-BCF-Smear), a decoupled two-part nonparametric estimator that fits a Probit BCF on the full sample to model participation and a Gaussian BCF strictly on the active subpopulation to model positive intensity. Subpopulation Propensity Adjustment (SPA) mitigates regularization-induced confounding within the active subpopulation, and Duan's non-parametric smearing estimator re-transforms predictions to the response scale, bypassing the fragile scale multipliers of parametric models. A comprehensive simulation benchmark shows that ZIC-BCF-Smear matches well-specified parametric models without their assumptions and remains stable and calibrated where competitors either diverge or pathologically over-cover for nonlinear data. The method is then applied to individual-participant data from the SPECTRUM randomized phase III head-and-neck cancer trial, estimating the causal effect of assignment to panitumumab plus chemotherapy on the number of days patients spend with severe toxicity, an outcome for which a quarter of patients contribute an exact zero. Panitumumab assignment raises that burden by about ten days per patient under the trial's own differential treatment duration, and the decoupled structure localizes the effect mainly in the positive-intensity part rather than in whether the reconstructed severe-AE burden is positive.

We develop a class of semiparametric zero-altered models (ZAMs) for count data, estimated through the generalized method of moments (GMM). The proposed framework extends the utility of ZAMs by allowing for both under- and over-dispersion while preserving support on the entire set of natural numbers. This offers an alternative to other models that can experience computational challenges (e.g. Conway-Maxwell-Poisson) or whose supports can be truncated as functions of the parameter space (e.g. generalized Poisson). By formulating estimation through moment restrictions rather than full likelihood specification, the GMM approach provides substantial robustness against model misspecification. In particular, inference based on likelihood methods for parametric generalized linear models (GLMs), such as those built on Poisson, negative binomial, or related count distributions, can become unreliable when the assumed distributional form is violated, often leading to biased estimates and invalid inference. In contrast, our methodology requires only the first two moments of the distribution, ensuring consistent estimation even when higher-order distributional features are misspecified. We establish the asymptotic properties of the estimators, including consistency and asymptotic normality, under general regularity conditions. The flexibility of the approach is illustrated through both simulated experiments and real-data case studies, demonstrating superior performance relative to popular parametric GLMs, particularly in settings characterized by distributional misspecification. To facilitate practical adoption, we provide accompanying R functions for easy implementation, enabling practitioners to apply the proposed methods in applied settings without requiring bespoke programming.

The two main traditional player statistics used in soccer are goals scored and assists, with an assist credited when a player's pass immediately leads to a teammate scoring. Attributing credit solely to the player who makes the final pass before a goal fails to account for the contributions of multiple players in the buildup to the goal. Consequently, traditional statistics provide only a limited view of player involvement. In this paper, we use multivariate Hawkes processes (MHPs) to model timestamped sequences of passes and shots, which form continuous-time networks. MHPs provide a branching mechanism where an event (a pass or shot) can be generated as a child of a previous event, which allows us to quantify each pass's contribution to a subsequent pass or shot. This approach distributes credit across multiple passes within an attacking sequence, making it applicable across a wide range of situations. Building on this framework, we propose expected goal involvements (xGI), a metric that measures a player's involvement in creating goal scoring opportunities based on their passes. This metric identifies players who play key roles in creating goal scoring opportunities even if they do not make the final pass, offering a more comprehensive measure of attacking involvement and player performance.

We study testing a random geometric graph with a fixed kernel against an Erdős-Rényi graph with the same edge density. Under the null hypothesis, the graph has $n$ vertices and independent edges with probability $p$. Under the alternative hypothesis, latent points $X_1,\ldots,X_n$ are independent and uniform on $\mathbb{S}^{d-1}$, and edges are conditionally independent with probabilities $K(\langle X_i,X_j\rangle)$, where $p=\mathbb{E}[K(\langle X_1,X_2\rangle)]$. The goal is to determine a critical dimension $d_\star = d_\star(n,K)$: the total variation distance between the two graph laws tends to one for $d\ll d_\star$ and to zero for $d\gg d_\star$. For dense graphs, [BDER16] established $d_\star\asymp n^3$ for hard threshold kernels, whereas [MWX26] proved $d_\star\asymp n^{3/4}$ for smooth kernels with $K'(0)\ne0$. We connect these results by characterizing the detection thresholds for kernels that need not be differentiable at zero. We also determine the thresholds for smooth kernels whose first nonzero derivative at zero has finite order greater than one. Throughout these classes, signed triangle or quadrilateral counts attain the optimal detection threshold. For certain nondifferentiable kernels, quadrilateral counts are optimal while triangle counts are suboptimal.

We study the two-sample estimation of the powered Sliced Wasserstein functional under projected $r$-th moment assumptions. Let $N=n\wedge m$, where $n$ and $m$ are two sample sizes. We establish the minimax rates in the noncritical regimes. When $p<r<2p$, the minimax risk is of order $N^{-(r-p)/r}$, while for $r>2p$ it is of order $N^{-1/2}$. In both cases, the natural plug-in estimator is minimax rate-optimal. At the endpoint $r=p$, the minimax risk is bounded away from zero, so uniformly consistent estimation is impossible. At the critical order $r=2p$, we obtain the exact worst-case rate of the plug-in estimator and a lower bound for the minimax risk, leaving a logarithmic gap between the lower and upper bounds. The exact minimax rate in this critical case remains open.